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Mirrors > Home > MPE Home > Th. List > cnmpt2nd | Structured version Visualization version GIF version |
Description: The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmpt21.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt21.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
Ref | Expression |
---|---|
cnmpt2nd | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo2nd 7080 | . . . . . 6 ⊢ 2nd :V–onto→V | |
2 | fofn 6030 | . . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 2nd Fn V |
4 | ssv 3588 | . . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V | |
5 | fnssres 5918 | . . . . 5 ⊢ ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)) | |
6 | 3, 4, 5 | mp2an 704 | . . . 4 ⊢ (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) |
7 | dffn5 6151 | . . . 4 ⊢ ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧))) | |
8 | 6, 7 | mpbi 219 | . . 3 ⊢ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) |
9 | fvres 6117 | . . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd ‘𝑧)) | |
10 | 9 | mpteq2ia 4668 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) |
11 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
13 | 11, 12 | op2ndd 7070 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (2nd ‘𝑧) = 𝑦) |
14 | 13 | mpt2mpt 6650 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) |
15 | 8, 10, 14 | 3eqtri 2636 | . 2 ⊢ (2nd ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) |
16 | cnmpt21.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
17 | cnmpt21.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
18 | tx2cn 21223 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) | |
19 | 16, 17, 18 | syl2anc 691 | . 2 ⊢ (𝜑 → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) |
20 | 15, 19 | syl5eqelr 2693 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 Fn wfn 5799 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 2nd c2nd 7058 TopOnctopon 20518 Cn ccn 20838 ×t ctx 21173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 df-tx 21175 |
This theorem is referenced by: cnmptcom 21291 xkofvcn 21297 cnmptk2 21299 txhmeo 21416 txswaphmeo 21418 ptunhmeo 21421 xkohmeo 21428 tgpsubcn 21704 istgp2 21705 oppgtmd 21711 prdstmdd 21737 dvrcn 21797 divcn 22479 cnrehmeo 22560 htpycom 22583 htpyco1 22585 htpycc 22587 reparphti 22605 pcohtpylem 22627 pcorevlem 22634 cxpcn 24286 vmcn 26938 dipcn 26959 mndpluscn 29300 cvxscon 30479 cvmlift2lem6 30544 |
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